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For example you have a cell or already a bunch of cells. Those cell(s) divide and after several week you have a grown organism, for example a human with limbs, several different organs etc. However, how do cells know where up, down, left, right etc. is? I know left and right can be defined clearly in physics, for example by the Wu experiment verifying parity violation really occur.

However how do cells define those directions? What are the mechanisms for them to grow into a certain direction and tell other cells to attain a certain property?

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You may want to look into "Evolutionary - Developmental Biology", which deals with how an embryo develops into a grown organism.

Thus, by combinatorial specifying the identity of particular body regions, Hox genes determine where limbs and other body segments will grow in a developing embryo or larva. A paragon of a toolbox gene is Pax6/eyeless, which controls eye formation in all animals. It has been found to produce eyes in mice and Drosophila, even if mouse Pax6/eyeless was expressed in Drosophila.

In short, there's a sequence of gene activations that progressively map body into different regions. At first, there are just 3 regions (head, body, tail), then these get broken down into more and more finely defined sections.

You can think of these sections as GPS coordinates for the body. A gene combination at a certain point means that a limb will grow there.

You can look into "Sonic Hedgehog" gene, which allowed scientists to produce fuzzy growth on a fruit fly head.

The most interesting thing is that every living creature is mapped/built using the same toolbox of genes. The genes that define where a fruit fly wing is also define where a human arm is.

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Embryonic cells "know" where they are relative to each other by chemical signals, same as in adults. These molecules are known as morphogens (specific examples include the sonic hedgehog and β-catenin). The amount of morphogen in a region of cells determines which gene gets turned on and thus what it develops into. And the amount varies by how far they are from the source of the morphogen. Though AFAIK, scientists don't fully understand their mechanisms yet (like how they are dispersed, how the different concentrations are kept rigidly, etc.).

As for arms, etc., the earliest cells are unspecialized (stem cells). They "define" which becomes what early on by how they are arranged. Remember that their "up, down, left, and right" is relative to them, not to their surroundings. The first multicellular hollow ball of stem cells, the blastula, separates into distinct germ layers during gastrulation - the ectoderm, mesoderm, endoderm. Each of these layers develop into something different upon further division. What develops after that is basically a worm, composed of a series of repeating "segments" (somites) with a distinct head and tail area. Somites are initially undifferentiated. The job of determining which segment becomes what falls on Hox genes. So that for example, one segment becomes the head, another becomes the arm and pectoral girdle, another becomes a rib+vertebra, etc. Hox genes are arranged in clusters within the DNA relative to which gets expressed first. The first hox gene is for the head for example, while the last is for the tip of the tail (the tipmost segment of the coccyx in humans). Though these clusters themselves are often repeated into a kind of redundant failsafes. This page from the University of Utah explains it pretty well. It helps if you view embryogenesis as more or less echoes of evolution in fast forward.

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I wanted to add some helpful references. The 6th edition of the Gilbert Developmental Biology textbook is available on NCBI bookshelf. It's a bit old (2000), but much of the information is still relevant. You can search this textbook for specific terms but not browse.

There is also a collaborative science/fashion project between the Storey sisters, called Primitive Streak, which documents the early stages of human development.

For any given developmental process, there are a number of genes being activated and/or inactivated at that stage that determine how and when structures grow. Events can determine axes too; for instance, the dorsal/ventral axis of the embryo is determine by how the embryo blastocyst implants in the uterine wall. The left-right axis of the body is determined by the direction that fluid travels through a structure called the node, which is directed by hair-like structures called cilia. If nodal flow is backwards, your internal organs will switch sides, called situs inversus.

Back to the limb example... in tetrapods, or animals with 4 limbs, the area where limbs emerge is based on Hox gene expression, and whether or not it will become a forelimb (arms) or hindlimb (legs) is determined by either Tbx5 or Tbx4 gene expression, respectively.

The proximal-distal axis (proximal = towards the trunk; distal = away from the trunk) is determined by FGF gene expression, which induces formation of a structure called the apical ectodermal ridge (AER).

The anterior-posterior axis (head to tail) is determined by Sonic hedgehog gene expression in a region called the zone of polarizing activity (ZPA).

The dorsal-ventral axis (back-front; or back of the hand/arm - palm/inside of arm) is determined by the Wnt7a gene being expressed on the dorsal side.

The outgrowth of the limb is determined by Hox gene expression.

Digit (finger) formation is dependent on cell death between the fingers - hands start webbed, kind of like ducks, and then genes such as the BMPs cause cells between the fingers to die.

Terms to look up for more detail, since I don't have enough street cred here to post more than 2 links yet:

  • Snapshot Summary: The Tetrapod Limb (Gilbert book)
  • apical ectodermal ridge (AER)
  • zone of polarizing activity (ZPA)
  • situs inversus
  • Hox genes
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  • $\begingroup$ That node-flow ciliar thing causing situs inversus is so amazing! $\endgroup$ Commented Sep 19, 2016 at 0:46
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yea the hox genes are essential for the the specification of the location but the actual growing of the limbs are determined by a series of protein and other factors. proximal limb contains fgf(fibroblast growth factors) wnt(look up wnt pathway) and high levels of retonic acid where distal limbs shows high fgf's high wnt and little to none RA. the sonic the hedgehog gene talked about in the other answer has to to do with the anteriorizing(up)/posteriozing(down) of the autopod(digits, carpals). as to dorsal ventral wnt would be the culprit

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    $\begingroup$ Can you please work on your answer and also add some references to it? At the moment it's hard to understand without basic knowledge of the field. $\endgroup$
    – Chris
    Commented Dec 10, 2014 at 6:42
  • $\begingroup$ I concur with @Chris: with some elaboration here and there and the necessary citations this could be a great answer $\endgroup$
    – AliceD
    Commented Dec 10, 2014 at 10:40
  • $\begingroup$ why would i have to cite topics that i have learned through school, it is a topic that i am familiar with as far as elaboration sure what exactly would you recommend i elaborate on. $\endgroup$
    – Mario Diaz
    Commented Dec 10, 2014 at 21:12

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